Impact of wind energy rotor wakes on fixed-wing aircraft and helicopters

IMPACT OF WIND ENERGY ROTOR WAKES ON FIXED-WING AIRCRAFT

AND HELICOPTERS

Berend G. van der Wall, berend.vanderwall@dlr.de, DLR (Germany)

Dietrich Fischenberg, dietrich.fischenberg@dlr.de, DLR (Germany)

Paul H. Lehmann, paul.lehmann@dlr.de, DLR (Germany)

Lennert B. van der Wall, l.vanderwall@tu-bs.de, TU Braunschweig (Germany)

Abstract

The wake of wind energy rotors is modeled as a tip vortex helix with a vortex strength estimated from its rotor thrust. A fixed-wing sail plane and a helicopter whose rotor is represented as a fixed-wing circular disk (in-stead of rotating blades) are subjected to the wake. In both cases the roll moment induced by the wake is compared to the maximum roll control moment of the aircraft. For comparison with revolving blades, the blade element momentum theory is applied to the isolated rotor and a simulation of an entire helicopter is used as well. It is found that typical on-shore power plants could be a hazard for sailplanes, but not for heli-copters. Large off-shore wind energy converters, however, could even be a danger for small helicopters that may be used for maintenance.

NOMENCLATURE

𝐴𝐴, 𝐵𝐵 Non-dimensional effective begin and end of rotor blade, referenced to 𝑅𝑅

𝐴𝐴 Rotor disk area, m² 𝑏𝑏 Wing span, m

𝑐𝑐, 𝑐𝑐𝑒𝑒𝑒𝑒 Airfoil chord and equivalent chord, m

𝐶𝐶𝑙𝑙, 𝐶𝐶𝐿𝐿 Blade element and aircraft lift coefficient

𝐶𝐶𝑙𝑙𝑙𝑙 Lift curve slope

𝐶𝐶𝑙𝑙,𝑊𝑊 Wake-induced roll moment coefficient

𝐶𝐶𝑇𝑇, 𝐶𝐶𝑇𝑇,𝑊𝑊𝑊𝑊 Thrust coefficients of helicopter rotor and

wind energy turbine

𝑐𝑐𝑛𝑛 Radial integral coefficients, 𝑛𝑛 = 0,1,2, …

𝑑𝑑0, 𝑑𝑑𝐶𝐶,

𝑑𝑑𝑆𝑆, 𝑑𝑑𝑊𝑊

Integral lift coefficients related to Θ0, Θ𝐶𝐶, Θ𝑆𝑆, 𝜆𝜆𝑊𝑊0

𝑒𝑒0, 𝑒𝑒𝐶𝐶,

𝑒𝑒𝑆𝑆, 𝑒𝑒𝑊𝑊

Integral moment coefficients related to Θ0, Θ𝐶𝐶, Θ𝑆𝑆, 𝜆𝜆𝑊𝑊0

𝑓𝑓𝑊𝑊 Spanwise lift weighing funciton

𝐿𝐿, 𝐿𝐿 Blade lift, N; non-dimensional blade lift 𝑀𝑀𝛽𝛽, 𝑀𝑀𝛽𝛽 Aerodynamic flap moment about the

flapping hinge, Nm; non-dimensional flap moment

𝑁𝑁𝑏𝑏 Number of rotor blades

𝑟𝑟 Non-dimensional radial coordinate 𝑅𝑅𝑐𝑐, 𝑟𝑟𝑐𝑐 Vortex core radius, m; non-dimensional

core radius, referenced to 𝑅𝑅 𝑅𝑅 Helicopter rotor radius, m 𝑅𝑅𝐶𝐶𝑅𝑅 Roll control ratio

𝑇𝑇, 𝑇𝑇𝑊𝑊𝑊𝑊 Thrust of the helicopter rotor and the

wind energy turbine, N

𝑢𝑢, 𝑣𝑣, 𝑤𝑤 Velocity components in 𝑥𝑥, 𝑦𝑦, 𝑧𝑧 directions, m/s

𝑈𝑈 Rotor blade tip speed, m/s

𝑣𝑣𝑖𝑖𝑊𝑊 Wake vortex induced velocity, m/s

𝑉𝑉𝑖𝑖𝑖𝑖 Induced velocity normal to rotor disk, m/s

𝑉𝑉𝑇𝑇, 𝑉𝑉𝑃𝑃 Non-dimensional velocities acting

tan-gential and normal at the blade element, referenced to 𝑈𝑈

𝑉𝑉𝑊𝑊 Wind speed, m/s

𝑉𝑉∞ Aircraft flight speed, m/s

𝑥𝑥, 𝑦𝑦, 𝑧𝑧 Hub-fixed coordinates, x pos. down-stream, y pos. starboard, z pos. up, m 𝑌𝑌0, 𝑦𝑦0 Vortex position within the rotor disk, m;

non-dimensional position, referenced to 𝑅𝑅

𝛼𝛼, 𝛼𝛼𝑊𝑊 Angle of attack, wake-induced angle of

attack, deg

𝛽𝛽𝑉𝑉 Core radius shape factor

Γ Wind turbine tip vortex circulation strength, m²/s

∆ Perturbation of a variable 𝛿𝛿𝑎𝑎 Aileron deflection, deg

Θ, Θ0,

Θ𝐶𝐶, Θ𝑆𝑆

Blade section pitch angle, collective, lateral and longitudinal control angle, deg Λ Wing aspect ratio

𝜆𝜆𝑖𝑖 Thrust-induced inflow velocity normal to

the rotor disk, non-dimensionalized by 𝑈𝑈 𝜆𝜆𝑊𝑊,

𝜆𝜆𝑊𝑊0

Wind turbine wake-induced inflow ratio and its amplitude, normal to the rotor disk, non-dimensionalized by 𝑈𝑈

𝜇𝜇 Rotor advance ratio, 𝜇𝜇 = 𝑉𝑉_∞cos 𝛼𝛼 𝑈𝑈⁄ 𝜇𝜇𝑖𝑖 Axial inflow ratio, 𝜇𝜇𝑖𝑖= −𝜇𝜇 sin 𝛼𝛼

𝜌𝜌 Air density, kg/m³

𝜎𝜎 Rotor solidity, 𝜎𝜎 = 𝑁𝑁_𝑏𝑏𝑐𝑐 (𝜋𝜋𝑅𝑅)⁄ 𝜓𝜓 Rotor blade azimuth, deg

𝜓𝜓𝑉𝑉 Wake age in terms of azimuth behind the

blade, deg

Ω, Ω𝑊𝑊𝑊𝑊 Rotor rotational speed of helicopter rotor

1. INTRODUCTION

The sequencing of take-off and landing at airports are governed by safety requirements that emerge largely from wake interaction hazards. The strong wake vortices of preceding aircraft can have severe consequences on the following aircraft, exceeding their control capability with fatal consequences. Not only large and heavy aircraft such as wide bodies like a Boeing 747 or Airbus A380 can generate such dangerous strong tip vortices, even smaller aircraft like the Antonov AN-2 with 5.5 tons maximum gross weight caused a fatal accident with a following smaller Robin DR400-180 vehicle of 1 ton gross weight due to wake vortex encounter [1].

Numerous investigations concerning the effects of rotorcraft encountering the wake vortex of fixed-wing aircraft have been conducted in the past decades. In the 1980s, NASA [2] and US Army [3] investigated medium sized helicopters flying through wake vorti-ces of airplanes using both flight testing and analyti-cal methods. A UH-1H helicopter trimmed at 60 kts was used to fly through the wake vortices of a Doug-las C-54 airplane at varying distances. Over the range of 0.42 nm to 6.64 nm, maximum rotor blade structural loads, helicopter attitude response and tail rotor flapping were measured. For these distances the helicopter reactions due to the wake were rec-ognized but did not constitute safety hazards. Numerical simulation was used in [4] and [5] to in-vestigate the effects from a pair of trailing vortices of a preceding large airplane on the flight dynamics of a fixed and a rotary wing aircraft. The responses of airplane and helicopter are described as different in a way that the helicopter reacts more damped to the disturbance by the tip vortex.

More recently, work has been conducted by the University of Liverpool and QinetiQ, [6]-[8]. They investigated the influence of an active runway from an international airport to helicopter operations from a nearby approach and takeoff area. For their work they used flight mechanics simulation tools to exam-ine the effects when a helicopter encounters the shed tip vortex from a large aircraft. A model of the vortex velocity profile was established by the use of LIDAR measurement data from the airport. Several calculations for the Lynx helicopter and forward speeds from hover to 80 kts were conducted. The results showed that the helicopter reaction is pri-marily dependent on the rotor position relative to the vortex center. In some combinations, hazardous helicopter reactions were recognized. The main question is, whether a rotorcraft which meets han-dling performance standards is able to recover the disturbed flight attitude after encountering the vortex. The number of wind energy (WE) power plants on the country side in Germany is huge and many are in notable proximity to airfields. Recently this

trig-gered investigation initially regarding the wake vor-tex hazard for sail planes encountering WE turbine wakes. The downstream wake of such horizontal axis WE turbines is characterized by a spiral helix of usually three blade tip vortices on the surface of the wake tube, as the number of blades is three for the overwhelming majority of the installed systems. This wake generates two different disturbances inside the tube and in the immediate vicinity of the tube itself. The tip vortex spiral on the surface of the tube is fed with circulation from the WE turbine blades and around each of these vortices induced swirl veloci-ties are generated that induce velociveloci-ties towards the turbine inside the tube – resulting in a global wind deficit – and adding on the wind velocity outside the tube.

Thus, inside the tube a global “wind deficit” is pre-sent that manifests itself as a loss of air momentum. Crossing the tube horizontally into its center will therefore generate a side-slip angle for the aircraft when penetrating the wake tube boundary on one side and this side-slip angle vanishes again when penetrating the wake tube boundary on the opposite side. At the boundary of the wake large horizontal vortex swirl velocities are encountered that change their sign at the boundary itself, representing a dual lateral pulse for the aircraft entering it.

The situation is very different when crossing the wake tube at its upper or lower boundary, i.e., in the immediate vicinity of the center of these vortices. In these cases the rotational swirl field of the vortices generates strong vertical velocities acting on the airplane’s wings. Depending on the aircraft size rela-tive to the wake vortex spacing as well as to the proximity to the vortex centers, one wing may be subjected to upwash and the other one to downwash at the same instance of time, generating a large roll moment. Here, two scenarios come into mind: one, where the fuselage center line hits the vortex axis center; and one, where the fuselage is in the middle between two successive vortex centers. In the for-mer case the vortex peak swirl velocities will be close to the middle of the fuselage with asymptotic decay towards the wing tips and in the latter case the peak swirl velocities affect the wing tips with decay towards the fuselage. Both cases can be con-sidered as potentially hazardous.

The WE turbine wake and the wake-rotor interac-tional problem are illustrated in Fig. 1, showing the staggered vortices of the wake spiral and a helicop-ter approaching it from the right (a). The inhelicop-teractional problem of a rotor passing the upper end of the wake spiral in almost normal direction to it can be treated in different ways. In a first simplified ap-proach the rotor may be viewed as a circular disk and handled like a wing of small aspect ratio, (b). In this case the turbine’s blade tip vortex has a station-ary position on the wing and generates upwash with

increased wing lift on left of its center and downwash with accordingly less lift on the right of it. A refined model with four individual rotor blades is shown in (c). In this case the problem is unsteady in general because the rotating blades enter and pass the tur-bine’s tip vortex induced velocity field during their revolution. Thus, the mathematical treatment is much more involved.

( a ) Sketch of a WE turbine’s wake with aircraft

( b ) Circular wing in WE turbine wake

( c ) Helicopter rotor in WE turbine wake

Fig. 1: Sketch of the WE turbine wake – rotor

inter-action problem.

2. TECHNICAL APPROACH

2.1. The wind turbines and the wake model

The investigations of this paper focus on two WE turbines of different power class: a representative on-shore 3 MW turbine and a representative off-shore 7 MW turbine. The reference chord at 93% radius is used to define the initial tip vortex core radius, while the equivalent solidity of the WE rotors is based on the thrust-weighted chord distribution. The helicopter investigated in this paper is the Bo105, representative for the 2-2.5 ton class. Data for the WE turbines and the helicopter are given in Table 1. The “worst case” scenario is of interest, which is the operational condition of the WE turbines

with maximum tip vortex circulation strength which can be estimated from the rotor thrust coefficient.

Table 1: Dimensions and properties of the helicopter

rotor and of the WE turbine rotors.

Rotor Bo105 3 MW 7 MW 𝑅𝑅, m 4.91 56.5 77.0 RPM 424 7-14 5-11 𝑉𝑉𝑊𝑊,Ω𝑚𝑚𝑖𝑖𝑛𝑛, m/s -/- 3-5 3-5 𝑉𝑉𝑊𝑊,Ω𝑚𝑚𝑎𝑎𝑚𝑚, m/s -/- 13 12 W, rad/s 44.4 0.733-1.466 0.524-1.152 𝑈𝑈, m/s 218.0 41.4-82.8 40.3-88.7 𝑐𝑐(0.9𝑅𝑅),m 0.270 1.000 1.363 𝑐𝑐𝑒𝑒𝑒𝑒, m 0.270 1.684 2.295 𝑁𝑁𝑏𝑏 4 3 3 σ 0.070 ≈ 0.0285 ≈ 0.0285 It must be mentioned here that the definition of a rotor thrust coefficient in WE terms is different from that used in the helicopter community. In WE terms the reference dynamic pressure is based on half of the air density and the wind speed, (ρ 2⁄ )𝑉𝑉_𝑊𝑊2, but in helicopter analysis the equivalent dynamic pressure is based on the air density and blade tip speed: ρ𝑈𝑈2.

(1) , 2 2 2 ,

( / 2)

1

2

WE WE T WE T W WE WE WE W T T WE WE

T

T

C

C

V A

A U

V

C

C

U

ρ

ρ

=

=





⇒

_{= }

_





The specific blade loading, defined as 𝐶𝐶_𝑇𝑇

⁄

𝜎𝜎, is an indicator in helicopter rotor analysis for the begin-ning of aerodynamic stall onset somewhere along the rotor blade when it exceeds a value of 0.12. From the definition of the specific blade loading the tip vortex strength can be estimated by means of lifting line theory as [9]:

(2)

₂ ,

2

2

T WE WE WE WE T b W T WE b WE

C

U

c

U

R

N

C

V

N

C

π

σ

π

Γ =

=

=

W

The thrust coefficients of WE turbines as a function of wind speed for the 3 MW turbine investigated here is shown by the dotted line in Fig. 2. Combined with the rotational speed of the turbines, the tip vor-tex circulation strength can be evaluated by Eq. (2) and is shown in Fig. 2 as solid line for the 3 MW turbine and as dashed line for the 7 MW turbine. It can be seen that a wind speed of 𝑉𝑉_𝑊𝑊= 10 m s⁄ pro-vides the largest value of tip vortex circulation, there-fore this condition is used in the analysis of helicop-ter rotor trim.

W V W x y z Γ Γ Γ Γ Γ Voo Γ Γ Γ y/R x/R Γy /R0 Voo y /R Γ Γ Γ y/R y, W x/R Γ 0 Voo

Fig. 2: WE turbine operational parameters and

re-sulting tip vortex circulation strength.

Using the value of 𝐶𝐶_{𝑇𝑇,𝑊𝑊𝑊𝑊}= 0.764 given (which corre-sponds to 𝐶𝐶_𝑇𝑇 = 0.00758 and 𝐶𝐶_𝑇𝑇⁄ = 0.266 in heli-𝜎𝜎 copter notation; this indicates that some portion of the blade is in a stalled condition) the peak circula-tion strength results in Γ = 63.7 m² s⁄ for the 3 MW turbine and to 76.4 m² s⁄ for the 7 MW turbine (as-suming 𝐶𝐶_{𝑇𝑇,𝑊𝑊𝑊𝑊} to be the same for both). It must be noted that the rotational speed as function of the wind speed is not revealed by the manufacturers and thus had to be estimated by the authors. Also, the 𝐶𝐶_{𝑇𝑇,𝑊𝑊𝑊𝑊} curve of the 7 MW turbine is unknown and therefore the same 𝐶𝐶_𝑇𝑇 as for the 3 MW turbine was used instead, leading to a circulation strength of 98.6 m² s⁄ .

The WE turbine wake-induced velocities are com-puted numerically at distances up to four WE rotor radii behind the turbine, and around distances cen-tered at 500 m behind it. The wake is represented by eight revolutions of each blade’s wake (beginning at the turbine in the first case, centered about the mean distance in the second case). Every revolution is discretized by 72 straight line vortex elements, each one representing a 5° increment. Induced ve-locities of these finite-length straight vortex ele-ments, including a core radius model, are computed numerically.

The swirl velocity profile includes a core radius 𝑅𝑅_𝑐𝑐, a lateral position within the rotor disk 𝑌𝑌₀ and its swirl velocity magnitude depends on the vortex circulation strength Γ. Coordinates 𝑥𝑥, 𝑦𝑦, 𝑧𝑧 and lengths such as the core radius and vortex location within the rotor disk are made non-dimensional by the helicopter rotor radius 𝑅𝑅, velocities are referred to the tip speed 𝑈𝑈 = Ω𝑅𝑅 to provide the wake-induced inflow ratio λ𝑊𝑊

as fraction of the helicopter tip speed. The circula-tion is made non-dimensional by division through 𝑈𝑈𝑅𝑅. For an analytical solution of the problem sketched in Fig. 1 (c) the WE turbine vortex within the rotor disk is replaced by an equivalent infinitely long straight line vortex as sketched in Fig. 3.

Fig. 3: Replacement of the WE rotor spiral vortex of

strength Γ by an infinite long straight line vortex with strength Γ_{𝑒𝑒𝑒𝑒} to represent the vortex-induced velocity field in the upmost point.

This straight line vortex is modeled with a core radi-us model of Burnham-Hallock [10], which is a spe-cial case of the Vatistas’ model [11]. The equivalent circulation Γ_{𝑒𝑒𝑒𝑒}, or the inflow ratio 𝜆𝜆_𝑊𝑊0, are then esti-mated based on the computed wake-induced veloci-ty profiles. It turns out that for the cases investigated here this equivalent circulation is about half of the value of the WE spiral vortex strength.

(3)

(

)

(

)

0 0 0 2 0 0 0 2 ₂ 2

2

sin

;

2

sin

eq iW W c eq W W c

v

y Y

U

_{U y Y}

r

y

UR

r

y

R

r

λ

π

y

λ

λ

π

y

=

+

Γ

−

=

−

Γ

−

=

=

−

+

The peak value of the inflow ratio is obtained at the core radius itself.

(4) _, 0

2

4

eq W W max c c

r

UR

λ

λ

π

Γ

=

=

Natural diffusion is represented by time-dependent decay (or aging) functions for both the circulation strength (which reduces asymptotically to zero for long time) and the vortex core radius (which widens with time) in the following manner, following [10] and [12]. (5)

(

)

6 0 0 0. 01932 0 2 0

5 10

1

V c c v c WE WE

rad s

e

R

r

R

R

R

y

y

− −

⋅

+

Γ = Γ

W

=

The initial core radius 𝑅𝑅_𝑐𝑐0 is set to 5% of the WE chord length at 93% radius (i.e., the reference chord at the blade tip area). Both decay factors of the core

0 20 40 60 80 0.0 0.5 1.0 1.5 0 5 10 15 20 25 3 MW 7 MW x y z Γ Γ Γ Γ eq eq

radius and the circulation aging functions are empiri-cal, but based on measurements. These ‘’aging’’ functions of the tip vortex circulation and core radius are shown in Fig. 4. For the wind speed of 10 m/s investigated here 40 revolutions of the WE turbine account for a horizontal distance of 17.9 WE rotor diameters (= 2 km) downstream in case of the 3 MW turbine and 15.6 WE rotor diameters (= 2.4 km) in case of the 7 MW turbine. It can be seen that the vortex core radii range between 1% of the helicopter rotor radius shortly behind the WE turbine to about 40% far away from it.

Fig. 4: Circulation and core radius aging functions.

For simplification the hypothesis is made that within the aircraft or helicopter rotor disk the vortex-induced velocity field does not change (i.e., an equivalent infinite long straight vortex is assumed). This is justified since the WE rotor radius is several times larger than the rotor radius. In case of the 3 MW turbine the WE-helicopter rotor ratio is 11.5 and in case of the 7 MW turbine it is 15.7. Therefore, the wake curvature within the rotor disk can be ig-nored.

Inflow ratio distributions that are computed based on Eq. (3) for arbitrarily chosen values of 𝑟𝑟_𝑐𝑐= 0.115 (solid line) and larger core radii (2, 3, 4, 5 times the value, dashed lines) are shown in Fig. 5 (a) for a fixed core position 𝑦𝑦₀= 0.3. In Fig. 5 (b) the vortex is centered in the hub and its core radius is larger than the rotor radius. It represents a cut through the rotor center in lateral direction and the WE vortex-induced velocity profile within it, having a lateral offset with respect to the rotor center as sketched in Fig. 1. In the first case Fig. 5 (a) the WE vortex-induced inflow profile is very non-linear, while in the second case (b) it is practically linear.

2.2. The fixed-wing aircraft model

For determination of vortex induced influences on an aircraft, the strip method has found widespread use as aerodynamic interaction model. It is based on lifting line theory and describes the additional aero-dynamic forces and moments acting on an aircraft in

( a ) 𝑟𝑟_𝑐𝑐< 1, 𝑦𝑦₀> 0

( b ) 𝑟𝑟_𝑐𝑐= 2, 𝑦𝑦₀= 0

Fig. 5: Sketch of different possible induced velocity

distributions across the rotor disk.

a spatial wind field, e.g. wake turbulence [1], [13], [14]. For computation of the forces and moments, the lift generating surfaces of the aircraft are subdi-vided into sections for which the vortex influence is determined. At each strip the additional angles of attack and angles of sideslip due to the local wind/vortices are computed. Using a suitable lift gradient, an additional lift is obtained for each strip. These local lift increments are weighted elliptically in span direction and then summarized, as well as the corresponding moments.

This method was validated against wind tunnel tests [15] and flight test data in [16] and [17]. The trans-versal flight of a fixed-wing aircraft into the tip vortex helix behind a wind energy rotor is considered to be a worst case scenario concerning yawing and rolling moment impact (Fig. 6). For a rotating-wing aircraft, any orientation of a vortex within the rotor will cause pitching and rolling moments. For an assumed flight path approximately parallel to the vortices axes, which is roughly in maximum and in minimum rotor tip altitude, the induced rolling moment is primarily of interest. Crossing the wake in shaft hub altitude, a yawing moment impact is dominating.

Fig. 6: Transversal flight into the rotor blade tip

vor-tex behind a wind energy rotor.

Whereas the yawing moment influence is of short duration, the rolling moment impact is stronger and of longer duration – depending on the aircraft air-speed and encounter scenario. The rolling moment can be computed particularly well with the strip method.

The method is applied for two aircraft flying in the wind turbine rotor wake: Fig. 7 (a) a sailplane (Ka8-b) with 15 m wingspan and flying at the airspeed of 𝑉𝑉 = 17 𝑚𝑚 𝑠𝑠⁄ , and Fig. 7 (b) a helicopter with a rotor disk diameter of 10 m (Bo-105) and an airspeed of 𝑉𝑉 = 20 𝑚𝑚 𝑠𝑠⁄ .

For computation of the vertical forces and rolling moments generated by the local up- and downwind of the WE turbine wake vortices, the wing of the encountering aircraft is subdivided typically into 16 strips, see Fig. 7. The additional local angle of attack ∆𝛼𝛼𝑊𝑊 induced by the local wake vertical wind 𝑤𝑤𝑊𝑊(𝑖𝑖) is

determined at each strip 𝑖𝑖.

(6)

( )

arctan

W

( )

W

w i

i

V

a

∞

∆

=

Then the additional local lift ∆𝐶𝐶_𝐿𝐿(𝑖𝑖) is calculated at each strip with the overall aircraft lift gradient 𝐶𝐶_{𝐿𝐿𝑙𝑙}. (7)

∆

C i

_L

( )

=

C

_La

∆

a

_W

(

i

)

f

_W

(

i

)

A weighting function 𝑓𝑓_𝑊𝑊(𝑖𝑖) is applied assuming an elliptically lift distribution along the span 𝑏𝑏 with the corresponding lever arms at each strip position 𝑦𝑦(𝑖𝑖). (8)

f

_W

(

i

)

4

sin

arcco

s

2 ( )

y i

b

π

−





=

_

_





( a ) Sailplane

( b ) Circular disk approximation

Fig. 7: Strip method: wing discretization.

The effect on the rolling moment is computed with the respective lever arms as well and summed up for all strips. The applied overall aerodynamic lift gradients 𝐶𝐶_{𝐿𝐿𝑙𝑙} for the sailplane wing and the helicop-ter rotor disk are dehelicop-termined applying the Helmbold equation, depending on the aspect ratio Λ of the lift generating surface [18]. (9) 2

2

2

4

L

C

_a

=

π

L

+

+ L

The Helmbold equation is in between the Prandtl and the Barrows formulation and applicable for high aspect ratio wings as well as low aspect ratio tail surfaces or rotors [18]. The overall lift gradient is determined to (a) 𝐶𝐶_{𝐿𝐿𝑙𝑙}= 6 for the sailplane with Λ = 15.9, and (b) 𝐶𝐶𝐿𝐿𝑙𝑙= 1.83 for the helicopter,

rep-resented by a circular disk with Λ = 4 𝜋𝜋⁄ = 1.273.

2.3. The helicopter model

For flight mechanics purposes the Institute of Flight Systems at DLR uses the non-linear “Helicopter Overall Simulation Tool” HOST for desktop simula-tion [19]. HOST was developed by Airbus Helicop-ters and is now used and further developed in coop-eration with ONERA and DLR. It is a modular tool that has the ability to simulate any type of helicopter and calculate trim, time domain response and per-form linearization.

For the results obtained here HOST was used in a special configuration with the “Atmospheric

Envi-ronment Submodule”. This submodule is connected to HOST’s flight mechanics model, calculates dis-turbed air and provides all relevant helicopter parts with the related turbulence parameters. This is main-ly the modified velocity due to the flow field of the vortex. An interaction which models the influence from rotor induced velocity to the vortex is not mod-elled. The Atmospheric Environment Module is able to simulate the following types of wake: atmospheric turbulence, big size aircraft vortex wake model, gusts in all directions, and the flow around different types of buildings.

All calculations performed here were conducted with the flight mechanics model of the BO105 helicopter and the so-called “Big Size Aircraft Vortex Wake Model”. Therein, the vortices are described following the Lamb-Oseen formula [20], [21]. Its equivalent circulation strength or swirl velocity 𝑉𝑉_𝑐𝑐, a core radius 𝑅𝑅𝑐𝑐 and a decay factor 𝛽𝛽𝑉𝑉 must be provided; they are

estimated from the computed WE wake-induced velocity field. At a distance 𝑦𝑦 = 0 from the core cen-ter this formula has a numerical singularity, but it can be proven that the analytical boundary value results in 𝑣𝑣_𝑊𝑊= 0, because the expression in brackets of Eq. (10) approaches the zero faster than the division by 𝑦𝑦. (10) 2

1

;

2

V c R y eq c W c c c

V

e

V

y

R

R

v

β

π

  −    





_Γ





=

−

=









Note that for 𝑦𝑦 = 𝑅𝑅_𝑐𝑐 the resulting swirl velocity is 𝑣𝑣𝑊𝑊= 𝑉𝑉𝑐𝑐�1 − 𝑒𝑒−𝛽𝛽𝑉𝑉� and for the value used of

𝛽𝛽𝑉𝑉= 0.97 the result is 𝑣𝑣𝑊𝑊≈ 0.63𝑉𝑉𝑐𝑐. Also, the

maxi-mum swirl velocity is obtained at 𝑦𝑦 ≈ 1.14𝑅𝑅_𝑐𝑐, i.e. 14% larger than the specified value of 𝑅𝑅_𝑐𝑐.

It is assumed that this wake model is able to repre-sent the idealized vortex in the wake of a WE turbine while it is provided with the required vortex data core radius and swirl speed at the core radius. Because a trim calculation is used to investigate the effects of the wake vortex (represented by an equivalent air-plane vortex) on the helicopter only the wake impact on the rotor is of interest. To achieve this no links between the wake model and helicopter parts other than the rotor are modelled. Due to the expected change in rotor thrust other changes to the aerody-namic helicopter parts are expected and should be visible in helicopter attitude.

The helicopter model used for these investigations, the Bo105, is a light utility rotorcraft and is used in maintenance of offshore wind farms. The model assumes an articulated rotor with rigid blades flap-ping about an effective hinge offset and the Meier-Drees model is used for the induced velocity calcula-tion. It was configured with a mass of 2200 kg.

The purpose of the complete helicopter trim analysis is to compare the results with the analytical model estimates in order to investigate the differences be-tween these two methods. For this comparison, the main rotor control angles dependent on the relative 𝑦𝑦-Positon from rotor to vortex center are used. For four scenarios A, B, C and D the helicopter reaction due to the vortex of a 3 MW or a 7 MW wind turbine in a distance of 𝑦𝑦 = 100 𝑚𝑚 or 500 𝑚𝑚 was computed. Therefore the HOST airplane wake model is provid-ed with the data given in Table 2 for these cases.

Table 2: Properties of the HOST big size aircraft

vortex wake model.

Scenario WE 𝑦𝑦, 𝑚𝑚 𝑅𝑅_𝑐𝑐, 𝑚𝑚 𝑉𝑉_𝑐𝑐, 𝑚𝑚 𝑠𝑠⁄ A 3 MW 100 0.393 10.5 B 3 MW 500 0.863 4.5 C 7 MW 100 0.542 12.0 D 7 MW 500 1.189 4.8

To determine the perturbation in main rotor control angles due to the aircraft vortex, the helicopter is trimmed with 10 km/h forward speed in the influence area of the vortex. The helicopter is positioned in a way that the vortex is in the same height as the rotor hub and is varied in position from 𝑦𝑦 = −2𝑅𝑅 to +2𝑅𝑅 in steps of 0.25𝑅𝑅. The perturbation control angles are obtained by subtracting the trim controls of the un-disturbed flow field without an aircraft vortex from the trim controls including the vortex.

2.4. The analytical rotor model

Based on the sketch of Fig. 1 (c) the velocities act-ing on a rotor blade element tangential (in the rota-tional plane, normal to the radial axis) and normal to the rotational plane can be established. It is as-sumed that the helicopter’s flight path is parallel to the WE turbine’s tip vortex axis, the vortex center lies in the plane of the rotor disk, and the rotor is horizontal. As indicated in the sketch the vortex cen-ter lies in the rotor disk and therefore only vertical vortex-induced velocities are acting on the blade elements. The vortex axis is assumed to be parallel to the rotor longitudinal 𝑥𝑥-axis and the vortex-induced velocities are a function of the lateral coor-dinate 𝑦𝑦 only. All velocities can be split into those components present in an isolated rotor (index 0) and those components due to the WE wake 𝜆𝜆_𝑊𝑊 from Eq. (3) that are considered as perturbations (Δ val-ues). (11) 0 0

sin

;

0

;

T T T T P z i W P P P W

r

V

V

V

V

V

V

V

V

µ

y

µ λ λ

λ

= +

=

+ ∆

∆

=

=

+ +

=

+ ∆

∆

=

A trim of the rotor requires collective and cyclic con-trols in order to establish the required thrust, propul-sive and lateral forces, as well as the hub moments needed for a steady flight. Any perturbations of the

velocities acting at the blades therefore require per-turbations in the controls in order to maintain the trim. Thus, the controls can be set up in a similar manner. (12) 0 0

sin

cos

sin

S C S

y

y

y

Θ = Θ + Θ

+ Θ

∆Θ = ∆Θ + ∆Θ

In a general case a ∆Θ_𝐶𝐶cos 𝜓𝜓 would be considered as well, but here a perturbation parallel to the x-axis is assumed (see Fig. 1 (c)) for simplification. This causes only lateral unbalance of disturbances while in longitudinal direction the disturbances are always balanced fore and aft of the rotor hub.

The blade section angle of attack is defined by three contributions: first, the pitch angle Θ, which is need-ed for the helicopter trim in undisturbneed-ed atmosphere. Second, a perturbation ΔΘ (Eq. (12)) to compensate the trim disturbance caused by the third contribution, namely the WE vortex-induced velocities given in Eq. (11). These perturbation control angles are computed by the requirement of keeping the trim constant.

First, this results into an equation for the steady (mean) lift perturbation to be zero. Second, the 1 𝑟𝑟𝑒𝑒𝑣𝑣⁄ aerodynamic flapping moment perturbation about the hub must be zero as well. Both of these values require a radial integration of the section lift distribution from the effective begin of the airfoiled part of the blade 𝐴𝐴 to the effective end of it 𝐵𝐵.

(13)

(

)

(

)

(

)(

)

(

)

2 0 2 2 0 0 2 ₂ 0

sin

sin

sin

sin

sin

sin

B B P T A A T B A B S _A B W A c

r

r

V

L

dL

V

dr

V

dr

dr

r

r

y

dr

r

y

r

µ

y

µ

y

y

µ

y

y

λ

y



∆



∆ =

∆

≈

_

∆Θ −

_





= ∆Θ

+∆Θ

+

−

−

−

+

+

+

∫

∫

∫

∫

∫

The equation for the moment is:

(14)

B A

M

β

r dL

∆

=

∫

∆

The wake integral of the perturbation lift poses prob-lems in analysis, since a Fourier series is needed for ease of further processing, but a broken rational function with periodic terms in both nominator and denominator is present. A Fourier analysis trans-forms this into the desired form.

(15)

(

)

(

)

(

)

0 2 ₂ 0 0 1

sin

sin

sin

cos

sin

B A c W Wi Wi i

r

y

r

dr

r

y

r

a

a

i

b

i

y

µ

y

y

y

y

∞ =

−

+

−

+

=

+

+

∫

∑

We need the mean value 𝑎𝑎_𝑊𝑊0 for keeping the mean value of the lift perturbation zero (∆𝐿𝐿���) and the 1 𝑟𝑟𝑒𝑒𝑣𝑣₀ ⁄ sine part 𝑏𝑏_𝑊𝑊1 for keeping the rotor roll moment per-turbation zero (in the rotating frame that is ∆𝑀𝑀����� =_{𝛽𝛽𝑆𝑆} 0), all higher harmonics are of no interest for the study here since they do not affect the rotor trim. A rotor pitch moment perturbation is not generated due to the vortex axis being parallel to the rotor 𝑦𝑦-axis (in the rotating frame: ∆𝑀𝑀����� = 0). _{𝛽𝛽𝐶𝐶}

2.4.1. Approximate solution for 𝒓𝒓_𝒄𝒄≫ 𝟏𝟏, 𝒚𝒚_𝟎𝟎= 𝟎𝟎

Let us first assume that the rotor is perfectly aligned with the vortex core center, i.e. 𝑦𝑦₀= 0, and that the core radius of the WE vortex is larger than the heli-copter rotor radius 𝑟𝑟_𝑐𝑐 ≫ 1 as in Fig. 5 (b). In this case the helicopter rotor experiences only an ap-proximately linear variation of vortex-induced veloci-ties laterally across the disk with zero velociveloci-ties in the center. Using Eq. (13) and inserting the linear form 𝜆𝜆_𝑊𝑊= 𝜆𝜆_𝑊𝑊0𝑦𝑦 𝑅𝑅⁄ = 𝜆𝜆_𝑊𝑊0𝑟𝑟 sin 𝜓𝜓 leads to

(16)

(

)

2 2 0 0 2 0 3 1 2 0 2 0 0 0 1

0

2

2

2

2

B A B S W S W S S W W i i B _i B A i _A A

r

dr

r

r dr

dr

c

c

c

c

d

d

B

A

r dr

i

d

c

µ

µ

µ

λ

µ

µ

λ µ

λ

−

= ∆Θ

+

+ ∆Θ

−

= ∆Θ

+

+

−

=

−

∆Θ

∆Θ

+ ∆Θ

−

=

_∫

=

∫

∫

∫

The application of the linear WE inflow variation to Eq. (14) leads to the second equation to determine the collective and cyclic perturbation controls.

(17)

(

)

2 0 2 3 3 0 2 2 0 0 0 3 2 4 0 0 4

0

2

3

3

8

4

4

4

S W S B W S S W W A B B A

r

A

r

r dr

r

dr

dr

c

c

e

e

e

c

c

µ

µ

_λ

µ

µ

λ

λ

∆Θ

∆Θ

+

−

= ∆Θ

+ ∆Θ

−

∆Θ

+ ∆Θ

−

=

+

+

=

∫

∫

∫

Now Eqs. (16) and (17) can easily be combined to solve for ΔΘ₀ and ΔΘ_𝑆𝑆.

(18) 0 0 0 0 0 0 0 0 0 0 W W S W S S W W S S W W S S

d e

e d

d e

e d

d

d

e

e

d

e

λ

λ

λ

−

∆Θ =

−

− ∆Θ

− ∆Θ

∆Θ =

=

An interesting result is obtained for the case 𝐴𝐴 = 0, 𝐵𝐵 = 1 and in hovering condition where 𝜇𝜇 = 0: (19)

∆Θ

_S

=

λ

_W0

and

∆Θ

₀

=

0

2.4.2. Exact solution for 𝒓𝒓_𝒄𝒄≫ 𝟏𝟏, 𝒚𝒚_𝟎𝟎= 𝟎𝟎

Following the derivation given in the appendix the coefficients

𝑎𝑎

_𝑊𝑊0 and

𝑏𝑏

_𝑊𝑊1 can be computed and the expressions equivalent to Eqs. (16) and (17) evalu-ated. The result is of course more involved. For the lift equation it is (20)

(

)

(

)

(

)

2 0 3 1 2 2 0 0 0 0

0

2

2

ln 1

1

/

S B W c A S S W W

c

c

c

d

d

r

d

r

µ

µ

λ µ

λ

= ∆Θ

−

=

−

+

+ ∆Θ

+

+

∆Θ

+ ∆Θ

For the moment equation it is

(21)

(

)

(

)

2 2 0 2 2 0 2 0 3 4 0 0 2

0

3

2

8

4

1

/

S B W c c A S S W W

c

c

c

r

r

c

e

r

e

e

µ

µ

λ

λ

= ∆Θ

+ ∆Θ





−

_

−

+

_





= ∆Θ

+ ∆Θ

−

+

To compute the perturbation angles of attack, Eq. (18) is used again.

2.4.3. Exact solution for arbitrary core radius and lateral vortex position

The most general solution of Eq. (13) is mathemati-cally very involved and the full derivation is given in the appendix. Here only the results of the wake inte-grals in Eqs. (13) and (14) are given. To simplify the expressions of the results the following abbrevia-tions are used.

(22)

ξ

=

r

2

−

y

₀ 2

+

r

_c2

η

=

2

y r

_{0 c} Then the wake integral of the lift equation becomes

(23) ₀

B

w W

A

d

=

∫

a

dr

With the derivation given in the appendix this

be-comes

(24) 2 2 2 0 2 2 0 2 2

ln

2

2

ln 1

2

B w A B c A B c A

d

y

r

r

y

µ

ξ

η

ξ

µ

ξ

η

ξ

ξ

η

ξ

=

+

+ +









+

_

+

_

+

+





+

+

+

and that of the moment equation becomes

(25)

( )

0 0 0 0 0 1 2 2 2 2 2 2 2 2 2 2 0 2 2

2

ln

2

2

2

ln 1

2

2 sgn

arctan

2

2

B W W A B A B c A B c A B c A B A

y

y

y

e

rb d

r

y

r

r

y

c

y

r

µ

ξ

η

ξ

µ

ξ

η

ξ

µ

ξ

η

ξ

ξ

η

ξ

ξ

η

ξ

=

=

+

+ +









+

+



₊

₊







+

+

+

−

+

+

+

+

−

+

∫

To compute the perturbation angles of attack, Eq. (18) is used again.

3. RESULTS

3.1. Wake-induced velocity fields

In order to get a feeling for the magnitude of WE turbine tip vortex induced velocities with respect to the helicopter rotor blade tip speed the vortex circu-lation is non-dimensionalized by Ω𝑅𝑅2 of the helicop-ter. This provides the order of magnitude of peak inflow ratio velocities with results shown for the val-ues of circulation given in Sect. 2.1. The parameters needed are summarized in Table 3.

At practical distances of 100-500 m to the WE tur-bine the core radius growth leads already to a signif-icant reduction of peak vortex induced velocities (shown later) that are in the range of the hovering rotor mean thrust-induced inflow ratio 𝜆𝜆_𝑖𝑖0= �𝐶𝐶_𝑇𝑇⁄ =2 0.0506, which is representative of a 2.3 ton Bo105 helicopter with a thrust coefficient of 𝐶𝐶_𝑇𝑇 = 0.00512.

Table 3: Circulation and peak induced inflow ratio. WE turbine 3 MW 7 MW Γ, m²/s (Eq. (2)) 63.7 98.6 Γ𝑒𝑒𝑒𝑒, m²/s (Eq. (3)) 31.9 49.3 𝜆𝜆𝑊𝑊0 (Eq. (3), 𝜓𝜓𝑉𝑉= 0°) 0.00474 0.00733 𝜆𝜆𝑊𝑊0 (𝑦𝑦 = 100𝑚𝑚) 0.00453 0.00704 𝜆𝜆𝑊𝑊0 (𝑦𝑦 = 500𝑚𝑚) 0.00400 0.00629 𝑟𝑟𝑐𝑐0 (𝜓𝜓𝑉𝑉= 0°) 0.0102 0.0138 𝑟𝑟𝑐𝑐0 (𝑦𝑦 = 100𝑚𝑚) 0.080 0.110 𝑟𝑟𝑐𝑐0 (𝑦𝑦 = 500𝑚𝑚) 0.177 0.242 𝜆𝜆𝑊𝑊,𝑚𝑚𝑎𝑎𝑚𝑚 (Eq. (4), 𝜓𝜓𝑉𝑉= 0°) 0.233 0.266 𝜆𝜆𝑊𝑊,𝑚𝑚𝑎𝑎𝑚𝑚 (𝑦𝑦 = 100𝑚𝑚) 0.0283 0.0320 𝜆𝜆𝑊𝑊,𝑚𝑚𝑎𝑎𝑚𝑚 (𝑦𝑦 = 500𝑚𝑚) 0.0113 0.0130

The peak inflow ratio at the vortex core radius is shown in Fig. 8. Instead of the time development as shown in Fig. 4 here the development is given in terms of the distance downstream the WE turbine, beginning with one rotor diameter behind the WE blade tip, i.e., the helicopter would have a clearance of only one rotor radius. Initially, the peak WE vor-tex-induced inflow ratios even exceed the hovering helicopter’s mean induced inflow value, but with increasing distance to the WE turbine they continu-ously decrease, following Eq.(5).

Fig. 8: WE vortex-induced peak inflow ratio.

For evaluation of the WE wake-induced velocities effect on rotor trim and the amount of trim controls needed for disturbance rejection, the induced veloci-ty field at various distances to the WE turbine need to be computed. The velocity profiles at the top cen-terline of the wake spiral are given in Fig. 9 (a) around a short distance to the turbine of 120 m and around a longer distance of 500 m in (b). The heli-copter’s size is sketched in the middle of the graph in order to provide the size of its rotor with respect to the spacing of the WE vortices.

It is evident that at this wind speed of maximum circulation strength the vortex separation is always larger than the helicopter rotor diameter. These

ve-locity data have been evaluated with a spatial reso-lution of 0.5 m in all directions. At young vortex ag-es, in the left half of Fig. 9 (a), some peak values of the 3 MW data are missed due to this resolution because the vortex core radii are smaller than the data resolution. However, within the rotor disk diam-eter of 9.82 m usually 19 samples are covered which is thought of as sufficient for the evaluation of the integral effect on the aerodynamic roll moment.

( a ) Short distance around 120 m

( b ) Medium distance around 500 m

( c ) Horizontal velocity profile at various distances

Fig. 9: WE vortex-induced velocity profiles at

vari-ous distances behind the WE turbine.

0.00 0.02 0.04 0.06 0.08 0.10 0 500 1000 1500 2000 2500 3 MW 7 MW hovering helicopter

Fig. 9 (c) shows the horizontal WE vortex-induced velocities along the vertical axis through a vortex center at different distances to the turbine. In all cases the 7 MW turbine’s velocities are larger than those of the 3 MW turbine, but the core radii are larger and the vortex spacing is larger as well.

3.2. Fixed-wing aircraft roll control ratio

In order to assess the potential severity of the wake impact on the encountering aircraft the induced roll-ing moment 𝐶𝐶_{𝑙𝑙,𝑊𝑊} is related to the controllability of the encountering aircraft applying the maximum roll control power 𝐶𝐶_𝑙𝑙�𝛿𝛿_{𝑎𝑎,𝑚𝑚𝑎𝑎𝑚𝑚}� [22], [23]. This relation de-fines the dimensionless vortex induced Roll Control Ratio RCR. (26)

(

)

(

)

, ,max , S,max

RCR =

R

(

CR =

)

(

)

l W l a l W l

C

fixed wing

rotorcr

C

C

C

aft

d

Θ

The maximum roll control power of the sailplane is assumed to �𝐶𝐶_𝑙𝑙�𝛿𝛿_{𝑎𝑎,𝑚𝑚𝑎𝑎𝑚𝑚}�� = 0.1, based on available data for corresponding aircraft types [24]. The max-imum helicopter roll control power is based on the assumption of an authority of 2500 𝑁𝑁𝑚𝑚 𝑑𝑑𝑒𝑒𝑑𝑑⁄ with a maximum longitudinal cyclic pitch angle of ΔΘ_{𝑆𝑆,𝑚𝑚𝑎𝑎𝑚𝑚}= 8 𝑑𝑑𝑒𝑒𝑑𝑑. It is thus determined for the equivalent circu-lar disk wing at a flight speed of 20 m/s as �𝐶𝐶𝑙𝑙�ΔΘ𝑆𝑆,𝑚𝑚𝑎𝑎𝑚𝑚�� = 0.22, which is more than twice as

high as for the sailplane. This is due to the fact that the sailplane aileron has a long lever arm, but only a relatively small control surface, while the helicopter uses the entire rotor blade as control surface.

Fig. 10 shows the magnitude of the roll control ratio of the Ka8-b sailplane with 15 m wing span and a helicopter with 10 m rotor disk diameter flying trans-versal 100 m behind the wind turbine across the upper part of the wake.

The results shown in Fig. 10 (a) indicate that a con-siderable rolling moment is imposed on the sailplane around the tip vortex position and between succes-sive vortices. Regions with RCR > 100% are ob-served, in which the rolling moment imposed from the WE turbine wake vortices cannot be compen-sated by the ailerons. The hazards during flight in a vortex center or in between vortices are in the same magnitude, but in opposite rotational direction. Shape and magnitude of the hazard zones depend not only on blade circulation and rotational speed, but also on aircraft wing span and wind speed. For the helicopter (10 m rotor diameter) flying at 20 m/s airspeed the situation is different (Fig. 10 (b)): the maximum roll control ratio never exceeds

RCR = 0.22 in the wake of a 3 MW turbine at 10 m/s wind speed. This is rated as fully controllable. More-over, noticeable rolling moments are only imposed at a flight into the center of a vortex, not in between two successive ones (as for the sailplane). The main reasons for this are higher roll control power of the helicopter and less influence to disturbances.

( a ) Sailplane with 15 m wing span

( b ) Helicopter with 10 m rotor disk diameter

Fig. 10: Distribution of RCR magnitude of a

sail-plane and a helicopter (figure center 100 m behind a 3 MW wind energy turbine), V_W = 10 m/s.

Fig. 11 shows the RCR results for a transversal heli-copter flight 100 m behind a 7 MW wind turbine at 10 m/s wind speed. The WE turbine rotor radius is 77 m, and the maximum roll control ratio is now RCR = 0.33. This is also rated as fully controllable.

Fig. 11: Distribution of RCR magnitude of a

helicop-ter (figure cenhelicop-ter 100 m behind a 7 MW wind energy turbine), V_W = 10 m/s.

3.3. Impact on helicopter trim

The following results were calculated with the heli-copter model described in Section 2.3. Fig. 12

dis-bl ade 1 bl ade 2 bl ade 3 bl ade 1 bl ade 2 bl ade 3 bl ade 1 bl ade 2 bl ade 3

plays the total induced velocity distribution within the rotor disk for scenario A of Table 2, including the disturbance due to a wing tip vortex. In this example the vortex is located at the same height as the hub but 0.5R left of the hub center. Positive values de-note downwash, thus the vortex is rotating clockwise when seen from behind with upwash on the left and downwash on the right of its axis. The fundamental induced velocity distribution represents a small lon-gitudinal gradient due to the small flight speed, but the lateral gradient is very small, following the Mei-jer-Drees model. The vortex axis could not be made fully parallel to the rotor x-axis and has a remaining 5° orientation misalignment, thus generating a slight aerodynamic pitch moment as well.

Fig. 12: Induced velocity distribution within the rotor

disk, 𝜇𝜇 = 0.0127, 𝐶𝐶_𝑇𝑇 = 0.0049.

Fig. 13 shows the perturbations of the main rotor trim control angles caused by the equivalent airplane blade tip vortex, dependent on the 𝑦𝑦-position of vor-tex core relative to the rotor hub, for each scenario of Table 2. The vortex location relative to the rotor hub ranges from 𝑦𝑦₀= −2 to +2, i.e. from the far left side to the far right side.

Fig. 13 (a) shows the collective rotor control angle perturbation. When the vortex is in the left half of the rotor (𝑦𝑦₀< 0) more collective is needed to compen-sate the vortex induced downwash that dominates over the rotor disk. When the vortex core is at the rotor hub (𝑦𝑦₀= 0) the collective control angle is nearly zero because the downwash on the left side equals the upwash on the right side. When the vor-tex is on the right side of the rotor less collective is needed because the vortex-induced upwash domi-nates over the rotor disk.

Comparing the graphs of the four scenarios the 7 MW turbine causes a larger perturbation in control angle than the 3 MW turbine. Also, the closer the distances to the turbine, the larger the control

per-turbation. This is obvious because the vortex core swirl speed has the largest values for these condi-tions.

( a ) Collective control

( b ) Longitudinal control

( c ) Lateral control

Fig. 13: WE vortex impact on helicopter flight trim, µ

= 0.0127, A = 0.25, B = 0.97.

Fig. 13 (b) displays the perturbations of the longitu-dinal rotor control angle Δθ_𝑆𝑆. Due to the vortex orien-tation relative to the rotor it was expected that aero-dynamic roll moments will appear causing changes in the longitudinal control angle to cancel the vortex impact on trim. The plot shows the biggest changes

when the vortex is located at 𝑦𝑦₀= −1, 0 and +1. When the vortex core is at the rotor hub (𝑦𝑦₀= 0) it causes as much downwash on the right side of the rotor as upwash on the left side. A positive aerody-namic roll moment appears and needs to be com-pensated by a positive longitudinal control angle. The vortex effect on the aerodynamic rotor roll mo-ment becomes zero for the positions 𝑦𝑦₀= ±0.63, which means that the downwash on the right side of the vortex and the upwash on the left are compen-sating each other with respect to the aerodynamic rotor roll moment.

When the vortex is located at either end of the rotor disk the longitudinal control needed to compensate its influence is negative in both cases. This is due to the vortex-induced velocity gradients within the rotor disk. For 𝑦𝑦₀= −1 the entire disk is immersed in downwash of the vortex, with largest values on the left side of the disk, thus a positive gradient from left to right. For 𝑦𝑦₀ = +1 the entire disk is immersed in upwash of the vortex, with largest values on the right side of the disk, thus again a positive gradient from left to right. Therefore, in both cases a negative lon-gitudinal control is needed to compensate this gradi-ent.

Finally, Fig. 13 (c) displays the perturbations of the lateral control angle Δθ_𝐶𝐶. From simple theory no perturbations are expected, since the vortex does not introduce a longitudinal gradient of velocities within the rotor disk. However, generally a perturba-tion appears with largest values between 𝑦𝑦₀= −1 and +0.5 (due to vortex misalignment and overall helicopter trim). All controls shown in Fig. 13 that are required to compensate the WE turbine wake vortex effects are small to moderate, compared to an avail-able control bandwidth of approximately 8 deg. As described in the technical approach the complete helicopter is simulated and not only the isolated rotor. Although no vortex effects on helicopter parts different from the rotor are considered the change in rotor thrust distribution and therefore its torque may cause a different tail rotor thrust. This represents a lateral force acting on the helicopter that has to be compensated by an associated lateral force of the main rotor for a trimmed flight. Due to the tail rotor position this change in tail rotor thrust will also cause a roll moment of the helicopter that the main rotor also has to compensate. Altogether, this leads to different helicopter pitch attitudes and roll angles, which in return will affect the other helicopter parts and finally will have an influence also on the lateral rotor control angle.

To confirm this hypothesis the helicopter pitch and roll angle perturbations relative to the trim without the WE vortex are shown in Fig. 14. The fuselage pitch angle (a) correlates with the lateral control angle perturbations of Fig. 13 (c) and the roll angle

in Fig. 14 (b) correlates to the collective control in Fig. 13 (a). Although this is no proof of the entire physical chain of events as outlined before, it is an indicator that the entire helicopter trim is modified by the vortex influence on the main rotor only.

In addition, the Bo105 is a hingeless rotor system with a relatively large equivalent flapping hinge, leading to a phase delay between control input to flapping reaction of roughly 78° (a central hinge as assumed in the analytical estimate has a delay of 90°). Therefore, a trim about the pitch axis mainly requires longitudinal control, but also a part of lateral control.

( a ) Pitch attitude perturbations

( b ) Roll angle perturbations

Fig. 14: Helicopter pitch and roll angle perturbations

due to a WE vortex.

3.4. Analytical estimation of rotor controls

First, the different velocity profiles resulting from different core radii are shown in Fig. 15 (a) and in (b) the variation of WE vortex induced velocities for varying vortex position within the rotor disk is shown for 𝑟𝑟_𝑐𝑐 = 0.5.

It is obvious that for small core radii a large non-linear impact on the rotor blade aerodynamics is present and only for core radii much larger than the rotor radius the velocity profile becomes practically

linear. Realistic WE vortex core radii will be a frac-tion of the rotor radius and then the vortex posifrac-tion within the rotor disk combines with the non-linearity of the velocity profile as seen in (b). The largest impact on rotor cyclic control is expected from these velocity profiles for the central WE vortex position where upwash is on the entire left rotor side and downwash on the entire right side, and also for the cases when the WE vortex is centered at the rotor radius when the WE induced velocity gradients across the disk appear largest.

The effect of the WE vortex lateral position within the rotor disk on longitudinal and collective control is shown in Fig. 16 (a) and (b) for different vortex core radii. Both trim control perturbations are linearly proportional to 𝜆𝜆_𝑊𝑊0 and therefore the ratio ΔΘ 𝜆𝜆⁄ _𝑊𝑊0 is independent of it; the magnitude of the controls is obtained when multiplying by the values actually encountered as given in Table 3.

( a ) Influence of vortex core radius, 𝑦𝑦₀= 0

( b ) Influence of vortex position, 𝑟𝑟_𝑐𝑐= 0.5

Fig. 15: WE vortex induced velocity profiles within

the rotor disk.

A potential vortex with a step jump from infinite up-wash to infinite downup-wash in its center is represent-ed by 𝑟𝑟_𝑐𝑐= 0, while realistic WE core radii are more in the range up to 𝑟𝑟_𝑐𝑐 = 0.1 − 0.5, depending on their age, see Table 3.

When the vortex center is to the left outside the rotor disk, 𝑦𝑦₀= −2, the entire rotor is immersed in the downwash side with diminishing magnitude towards the starboard (advancing) side. Therefore, the mean value is downwash with a gradient from left to right. This mean value requires a small positive collective (Fig. 16 (b)) to compensate the loss of thrust. The opposite is the case for a vortex position to the right outside the rotor disk, 𝑦𝑦₀= 2. For 𝑦𝑦₀= −2 the lateral gradient with more downwash of the vortex-induced velocities on the left side of the disk than on the right requires a small negative longitudinal cyclic (Fig. 16 (a)) to compensate the aerodynamic moment. The flat lines in the center area of the curves for 𝑟𝑟𝑐𝑐= 0 are due to the root cutout of A = 0.25.

When the vortex core reaches the left end of the rotor disk, 𝑦𝑦₀= −1, the maximum values of mean downwash and downwash gradients are obtained within the rotor, thus the largest amount of collective and cyclic are needed to compensate for the loss of lift and the large aerodynamic moments developing. The opposite is the case for a vortex position to the right outside the rotor disk, 𝑦𝑦₀= 1.

( a ) Longitudinal cyclic

( b ) Collective control

Fig. 16: Exact solution of the interactional problem

Any position of the WE vortex inside the disk com-bines downwash on its right side with upwash on its left until the center position 𝑦𝑦₀= 0 is a perfect bal-ance of both. In this case no collective is needed because the mean inflow is zero due to this balance. However, this represents the largest aerodynamic moment induced by the WE vortex and thus the largest cyclic control is needed in this case to coun-teract this moment.

A comparison with the complete helicopter flight trim results as shown in Sect. 3.3 with the analytical re-sults are given in Fig. 17. The advance ratio in both cases is 𝜇𝜇 = 0.0127, the peak collective control val-ues are taken as mean value of the extremes and the longitudinal control angles are taken from the vortex middle position at 𝑦𝑦₀= 0. Although the trend appears to be quite similar as already seen in Fig. 13 (a) and (b) the absolute values of the flight trim are larger by roughly a factor of 2.

Fig. 17: Comparison of HOST complete helicopter

trim perturbations with isolated rotor analysis, µ = 0.0127, A = 0.25, B = 0.97.

A reason for these differences is seen in the modi-fied trim of the entire helicopter. Any inclusion of the WE vortex modifies the main rotor torque that has to be compensated by the tail rotor with additional thrust, leading to a lateral force and a roll moment which both the main rotor has to compensate. This leads to different pitch and roll attitude of the entire helicopter which in reverse affects the cyclic and collective pitch of the main rotor blades. However, the trend is captured correctly by the analytical model.

The influence of the advance ratio 𝜇𝜇 is given next in Fig. 18 for the same variations of vortex position and core radius. Mainly the increase of dynamic pres-sure on the advancing (starboard) side of the rotor dominates over the loss of dynamic pressure on the retreating side.

With increasing 𝜇𝜇 the resulting curves of cyclic and collective control become non-symmetric and the control magnitudes are larger when the vortex is

located on the advancing side. Finally, the influence of the aerodynamically effective blade area can be analyzed.

So far, the parameters for a realistic rotor blade were used, i.e., an effective non-dimensional begin of the airfoiled part of the blade at 𝐴𝐴 = 0.25 and an effective blade tip at 𝐵𝐵 = 0.97. In Fig. 19 (a) and (b) a rotor blade beginning in the rotor center 𝐴𝐴 = 0 and ending at the true radius 𝐵𝐵 = 1 is shown for compar-ison with the former results and (c) and (d) give re-sults obtained with 𝐴𝐴 = 0.5 and 𝐵𝐵 = 0.97.

( a ) Longitudinal cyclic, 𝜇𝜇 = 0.3

( b ) Collective control, 𝜇𝜇 = 0.3

( d ) Collective control, 𝜇𝜇 = 0.5

Fig. 18: Influence of the advance ratio on the

inter-actional problem, A = 0.25, B = 0.97.

Mainly the root cutout dominates the magnitude of collective and cyclic control needed. This is espe-cially the case for small vortex core radii and vortex positions around the rotor center. In that case the root cutout effectively eliminates the vortex influence in this area. For core radii larger than the root cutout there is no practical difference to the results shown before in Fig. 16.

( a ) Longitudinal cyclic, 𝐴𝐴 = 0, 𝐵𝐵 = 1

( b ) Collective control, 𝐴𝐴 = 0, 𝐵𝐵 = 1

( c ) Longitudinal cyclic, 𝐴𝐴 = 0.5, 𝐵𝐵 = 0.97

( d ) Collective control, 𝐴𝐴 = 0.5, 𝐵𝐵 = 0.97

Fig. 19: Influence of the effective blade length on

the interactional problem, µ = 0.

4. COMPARISON OF RCR RESULTS

Assuming a control margin of ΔΘ_{𝑚𝑚𝑎𝑎𝑚𝑚}= 8° available (this is arbitrary to some degree and may be differ-ent for any individual helicopter) for compensating WE vortex-induced perturbations the peak values of Fig. 13 can be used to compute a roll control ratio RCR. The roll control ratio can here be defined as the combination of the perturbations in collective and cyclic controls, referenced to the maximum available control margin: (27) 2 2 0 S C max

RCR

∆

θ

+

∆Θ + ∆Θ

∆

=

Θ

This can be compared with the RCRs of Fig. 10 and Fig. 11 for the helicopter treated as a fixed-wing circular disk, and to the RCR computed from the perturbations of the simple analytical isolated rotor model given in Fig. 17. The result is given in Fig. 20 (a) for the variation of WE vortex position relative to the hub center, based on the data shown in Fig. 13. The largest combined collective and cyclic perturba-tions are needed when the vortex center is located at the right or left end of the rotor disk. This is

be-cause of the relatively large collective required at these vortex positions, leading to the entire rotor being exposed to the vortex downwash or upwash, depending on its position at the right or left end of the disk. This also generates the largest lateral in-duced velocity gradient across the disk, and thus the largest longitudinal cyclic control ΔΘ_𝑆𝑆 to compensate it.

( a ) Roll control ratio computed from Fig. 13

( b ) Maximum roll control ratios for the different methods of analysis

Fig. 20: Roll control ratio comparison of Bo105

heli-copter trim and analytical estimate.

Comparing Fig. 20 (a) with the results shown in Fig. 10 and Fig. 11, it can be observed that the maximum RCR of the helicopter rotor is found when the vortex is placed at the outer ends of the rotor disk. When the rotor is handled as a circular fixed-wing, the maximum RCR is obtained when the vortex is locat-ed in the middle of the wing. This is because the fixed-wing experiences a constant dynamic pressure all along span, while a rotating blade experiences a dynamic pressure quadratically increasing towards the blade tips. Therefore, when the vortex core is at the blade tip, its largest induced velocities are in the vicinity of the blade tips as well, and cause larger variation of lift than a vortex position at the hub cen-ter, where the dynamic pressure and thus resulting forces are very small.

In Fig. 20 (b) the peak values of these four scenarios shown in (a) are then compared to the maximum values of the results shown in Fig. 10 and Fig. 11 for the same distance relative to the WE turbine. First, the circular wing analysis compares surprisingly well with the HOST complete helicopter trim, despite the fact that the fixed-wing aerodynamic treatment is only based on the assumed flight speed of the wing of 𝑉𝑉 = 20 𝑚𝑚 𝑠𝑠⁄ , while the helicopter trim with rotating blades was performed near hover with 𝑉𝑉 = 2.5 𝑚𝑚 𝑠𝑠⁄ . It needs to be checked whether this agreement is accidentally or similar for other flight speeds as well. Second, the HOST RCR is larger than that of the analytical isolated rotor estimate. This is due to the larger collective computed by HOST and also due to the lateral cyclic predicted by HOST, which is zero in the simplified analysis due to the central flapping hinge. The longitudinal control required to compen-sate the WE vortex effects is found similar between HOST and the simple analysis.

The trend with respect to the strength of the vortex is predicted the same for all three methods: the stronger the vortex, the larger the RCR. In any of the four cases computed, the roll control ratio is found to be 0.25 in maximum and thus it is no problem for the helicopter to compensate the WE vortex effects, at least in steady trim.

5. CONCLUSIONS

In this paper the effect of a wind energy turbine wake vortex on the roll control ratio of a sailplane, a helicopter represented as a circular fixed-wing air-craft, a complete helicopter simulated by HOST, and a simplified analytical treatment of an isolated heli-copter rotor are compared. The major conclusions are:

• In 100 m distance to a 3 MW turbine a sailplane flying with a speed of 17 m/s would exceed an RCR of 1 and thus become uncontrollable, be-cause the vortex-induced roll moments cannot be compensated by the controls.

• In the same scenario, a helicopter of Bo105 size represented as a circular fixed-wing aircraft flying at a speed of 20 m/s would not be endan-gered due to an RCR of about 0.2.

• The complete Bo105 helicopter simulation with rotating blades, fuselage, tail rotor etc. at a flight speed near hover computes the RCR to 0.15, leaving even more margins for control. It is found that the main rotor controls are affected directly by the WE vortex, but also indirectly by an overall affected trim of the entire helicopter. • The simplified isolated rotor analysis predicts

only half of the RCR compared to the complete helicopter simulation. This is partly due to the missing fuselage, tail rotor etc. and their influ-ence on the overall trim.